Question

Rezolvați în z ecuația xy+3x-y=18​

Answer 1

Răspuns:

Explicație pas cu pas:

xy-y+3x=18   (scadem un 3)

y(x-1)+3(x-1)=15

(y+3)(x-1)=15

cazul 1.

y+3=5 => y=2

x-1=3 => x=4

cazul 2

y+3=3 =>y=0

x-1=5=>x=6

cazul 3

y+3=-5=> y=-8

x-1=-3=>x=-2

cazul 4

y+3=-3 => y=-6

x-1=-5=>x=-4

Answer 2

$\it xy+3x-y=18 \Rightarrow x(y+3) = 18 + y \Rightarrow x=\dfrac{18+y}{y+3}|_{-1} \Rightarrow \\ \\ \\ \Rightarrow x-1=\dfrac{18+y}{y+3}-1 \Rightarrow x-1=\dfrac{18+y-y-3}{y+3} \Rightarrow x-1=\dfrac{15}{y+3}\ \ \ \ (*) \\ \\ \\ x-1\in\mathbb{Z}\ \stackrel{(*)}{\Longrightarrow}\ y+3\in D_{15} \Rightarrow y+3\in\{\pm1,\ \pm3,\ \pm5,\ \pm15\} \Rightarrow$

$\it \Rightarrow y+3\in\{-15,\ -5,\ -3,\ -1,\ 1,\ 3,\ 5,\ 15\}|_{-3} \Rightarrow\\ \\ \Rightarrow y\in\{-18,\ -8,\ -6,\ -4,\ -2,\ 0,\ 2,\ 12\}$

$\it y+3\in\{-15,\ -5,\ -3,\ -1,\ 1,\ 3,\ 5,\ 15\} \Rightarrow\\ \\ \Rightarrow \dfrac{15}{y+3}\in\{-1,-3,-5,-15,\ 15,\ 5,\ 3,\ 1\}\\ \\ \\ (*) \Rightarrow x-1\in\{-1,-3,-5,-15,\ 15,\ 5,\ 3,\ 1\}_{+1} \Rightarrow x\in\{0,-2,-4,-14,\ 16,\ 6,\ 4,\ 2\}$

$\it S=\{(0,\ -18),\ (-2,\ -8),\ (-4,\ -6),\ (-14,\ -4),\ (16,\ -2),\ (6,\ 0),\ (4,\ 2),\ (2,\ 12)\}$